Lecture 5: Positive Definite and Semidefinite Matrices
Description
In this lecture, Professor Strang continues reviewing key matrices, such as positive definite and semidefinite matrices. This lecture concludes his review of the highlights of linear algebra.
Summary
All eigenvalues of S are positive.
Energy x_T_Sx is positive for x \(\neq 0\).
All pivots are positive S = A_T_A with independent columns in A.
All leading determinants are positive 5 EQUIVALENT TESTS.
Second derivative matrix is positive definite at a minimum point.
Semidefinite allows zero evalues/energy/pivots/determinants.
Related section in textbook: I.7
Instructor: Prof. Gilbert Strang
Problems for Lecture 5
From textbook Section I.7
3. For which numbers \(b\) and \(c\) are these matrices positive definite?
$$S = \left[\begin{matrix}1 & b\\ b & 9\end{matrix}\right] \hspace{12pt} S = \left[\begin{matrix}2 & 4\\ 4 & c\end{matrix}\right] \hspace{12pt} S = \left[\begin{matrix}c & b\\ b & c\end{matrix}\right] . $$
With the pivots in \(D\) and multiplier in \(L\), factor each \(A\) into \(LDL\)T.
14. Find the 3 by 3 matrix \(S\) and its pivots, rank, eigenvalues, and determinant:
$$\left[\begin{matrix}x_1&x_2&x_3\end{matrix}\right] \left[\begin{matrix} & & \\ &S& \\ & & \end{matrix}\right] \left[\begin{matrix}x_1\\x_2\\x_3\end{matrix}\right] = 4(x_1 - x_2 + 2x_3)^2 .$$
15. Compute the three upper left determinants of \(S\) to establish positive definiteness. Verify that their ratios give the second and third pivots.
$$ \textbf{Pivots} = \textbf{ratios of determinants} \hspace{12pt} S = \left[\begin{matrix} 2& 2& 0\\\ 2&5&3 \\\ 0&3&8 \end{matrix}\right] .$$
